Bi-shadowing of Quasi-semi-hyperbolic Pseudo-orbit

Abstract: In this paper, we introduce the concept of quasi-semi hyperbolic pseudo-orbits and prove that quasi-semi hyperbolicity implies quasi hyperbolicity provided the error magnitude are sufficiently small. We also have successively demonstrated that both finite quasi-hyperbolic pseudo-orbits and infinite quasi-semi hyperbolic pseudo-orbits possess the bi-shadowing property, and thus we establish the periodicity.

• The paper establishes that quasi-semi-hyperbolic pseudo-orbits, under small error conditions, imply a robust bi-shadowing property akin to quasi-hyperbolic behavior.
• The methodology utilizes contraction mapping, graph transforms, and tailored norm engineering to construct invariant subbundles and control error estimates.
• The findings have practical implications for reliable numerical simulations and the structural analysis of weakly hyperbolic dynamical systems.

Bi-Shadowing of Quasi-Semi-Hyperbolic Pseudo-Orbits

Introduction and Context

The study addresses a foundational problem in differentiable dynamical systems: the structure and stability of orbits under perturbation, particularly via shadowing properties. Classical shadowing—a pseudoorbit being closely tracked (“shadowed”) by an exact orbit—serves as the backbone of numerical simulation validity and qualitative dynamical analyses. This work generalizes the shadowing paradigm for quasi-semi-hyperbolic pseudo-orbits, a class interpolating between traditional quasi-hyperbolic and semi-hyperbolic behaviors, with a focus on bi-shadowing—where the shadowed orbit persists under both forward and backward dynamics. This departs from standard autonomous shadowing theory, extending to more nuanced, possibly non-autonomous, and weakly hyperbolic settings, where classical tools fail.

Main Definitions and Theoretical Contributions

The paper rigorously formalizes several notions:

• Quasi-Semi-Hyperbolic Pseudo-Orbits: Orbit segments admitting a non-invariant but continuously varying splitting, with dynamics exhibiting controlled expansion/contraction parameters ($\lambda, \varepsilon$), alongside off-diagonal "error" bounds.
• Bi-Shadowing Property: The existence of a real orbit for a perturbed system that shadows a given pseudo-orbit both forwards and backwards, within a prescribed error $\varepsilon_1$.
• Periodic Bi-Shadowing: For periodic pseudo-orbits, the shadowing orbit can also be taken to be periodic with the same period.

The central technical result is: For sufficiently small error parameters, any quasi-semi-hyperbolic pseudo-orbit is, after possibly adjusting the splitting, a quasi-hyperbolic pseudo-orbit. This demonstrates that the weaker quasi-semi-hyperbolicity, under small errors, implies strong, structurally stable hyperbolic-like behavior, ensuring applicability of shadowing results.

Proof Structure and Methodology

The proofs leverage advanced techniques from nonlinear functional analysis and hyperbolic dynamics:

• Contraction Mapping and Graph Transform Methods: The construction of invariant subbundles for the quasi-semi-hyperbolic sequence is achieved using fixed points of associated bundle maps, paralleling classical graph transform methods in hyperbolic theory. Contraction principles are employed in controlled normed spaces, with fine perturbation estimates ensuring uniqueness and persistence of relevant structures.
• Norm Engineering and Adapted Sequence Construction: By crafting “well-adapted” balance sequences, the expansion and contraction estimates are re-expressed in renormalized norms, allowing block-diagonalization arguments required for the application of shadowing principles.
• Limiting and Periodic Arguments: For infinite pseudo-orbits, diagonalization over increasing intervals yields convergence of shadowing points, using compactness of the manifold. The periodic case adapts fixed point formulations to periodic boundary conditions, guaranteeing periodicity of the shadowing orbit.

Main Results

In summary, the critical assertions can be organized as follows:

• Quasi-Semi-Hyperbolicity $\Rightarrow$ Quasi-Hyperbolicity: Under explicit smallness conditions for the block off-diagonal and transition errors, any $(\lambda, \varepsilon, \delta)$-quasi-semi-hyperbolic pseudo-orbit admits a splitting making it $(\widetilde{\lambda}, \delta)$-quasi-hyperbolic.
• Bi-Shadowing for Finite Pseudo-Orbits: For any sufficiently small $(\lambda, \varepsilon, \delta)$-quasi-semi-hyperbolic pseudo-orbit and any sufficiently close $C^1$ perturbation $g$ of $f$, there exists a true $g$-orbit that $\varepsilon_1$0-shadows the pseudo-orbit bi-directionally.
• Bi-Shadowing for Infinite Pseudo-Orbits: The bi-shadowing property extends via projective limits, ensuring global shadowing for bi-infinite pseudo-orbits.
• Periodic Bi-Shadowing: Any periodic quasi-semi-hyperbolic pseudo-orbit admits a periodic shadowing point for the perturbed system, with matched period and prescribed accuracy.

These results are derived with fully quantitative control over the constants and rely on explicit, verifiable hypotheses, enabling their practical application in further dynamical studies.

Implications and Future Directions

By establishing that quasi-semi-hyperbolic structures suffice for robust bi-shadowing under precise smallness assumptions, the work extends shadowing theory from strictly hyperbolic and semi-hyperbolic systems to a more versatile and broadly applicable class. The theoretical consequences are several:

• Numerical Simulations: The strong bi-shadowing property under weak hyperbolicity and small error regimes justifies the validity of long-term numerical integrations for these systems, even when hyperbolicity is non-uniform or partially broken.
• Robustness Under Perturbations: The results provide structural tools for handling non-autonomous and weakly non-uniform systems, crucial in applied contexts where strict hyperbolicity is rare.
• Invariant Measures and Symbolic Coding: The existence of periodic bi-shadowed orbits under minimal hyperbolic structure strengthens the foundation for symbolic coding, entropy computations, and statistical properties in general differentiable dynamics.

Future research directions prompted by this work might include:

• Extending to flows with singularities (beyond diffeomorphisms), where the interplay with continuous-time analogues is nontrivial.
• Analysis of stochastic or random dynamical systems where quasi-semi-hyperbolic pseudo-orbits naturally arise.
• Exploring finer statistical properties (SRB measures, large deviations) in the presence of quasi-semi-hyperbolic bi-shadowing.

Conclusion

This research rigorously builds a bridge between the quasi-semi-hyperbolic and quasi-hyperbolic regimes, establishing bi-shadowing properties in both finite and infinite horizon settings, including the periodic case. By developing a robust functional analytic framework, the results significantly extend the applicability of shadowing theory, laying groundwork for further advances in the qualitative and quantitative understanding of weakly hyperbolic dynamical systems.